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We have pointed out that a set representing a real situation is not an isolated collection. Sets, in general, overlaps with each other. It is primarily because a set is defined on few characteristics, whereas elements generally can possess many characteristics. Unlike union, which includes all elements from two sets, the intersection between two sets includes only common elements.

Intersection of two sets
The intersection of sets “A” and “B” is the set of all elements common to both “A” and “B”.

The use of word “and” between two sets in defining an intersection is quite significant. Compare it with the definition of union. We used the word “or” between two sets. Pondering on these two words, while deciding membership of union or intersection, is helpful in application situation.

The intersection operation is denoted by the symbol, " ". We can write intersection in set builder form as :

Intersection of two sets

The intersection set consists of elements common to two sets.

A B = { x : x A a n d x B }

Again note use of the word “and” in set builder qualification. We can read this as “x” is an element, which belongs to set “A” and set “B”. Hence, it means that “x” belongs to both “A” and “B”.

In order to understand the operation, let us consider the earlier example again,

A = { 1,2,3,4,5,6 }

B = { 4,5,6,7,8 }

Then,

A B = { 4,5,6 }

On Venn diagram, an intersection is the region intersected by circles, which represent two sets.

Intersection of two sets

The intersection set consists of elements common to two sets.

Interpretation of intersection set

Let us examine the defining set of intersection :

A B = { x : x A a n d x B }

We consider an arbitrary element, say “x”, of the intersection set. Then, we interpret the conditional meaning as :

I f x A B x A a n d x B .

The conditional statement is true in opposite direction as well. Hence,

I f x A a n d x B x A B .

We summarize two statements with two ways arrow as :

x A B x A a n d x B

In addition to two ways relation, there is an interesting aspect of intersection. Intersection is subset of either of two sets. From Venn diagram, it is clear that :

Intersection of two sets

The intersection set consists of elements common to two sets.

A B A

and

A B B

Intersection with a subset

Since all elements of a subset is present in the set, it emerges that intersection with subset is subset. Hence, if “A” is subset of set “B”, then :

B A = A

Intersection of disjoint sets

If no element is common to two sets “A” and “B” , then the resulting intersection is an empty set :

A B = φ

In that case, two sets “A” and “B” are “disjoint” sets.

Multiple intersections

If A 1 , A 2 , A 3 , , A n is a finite family of sets, then their intersections one after another is denoted as :

A 1 A 2 A 3 . A n

Important results

In this section we shall discuss some of the important characteristics/ deductions for the intersection operation.

Idempotent law

The intersection of a set with itself is the set itself.

A A = A

This is because intersection is a set of common elements. Here, all elements of a set is common with itself. The resulting intersection, therefore, is set itself.

Identity law

The intersection with universal set yields the set itself. Hence, universal set functions as the identity of the intersection operator.

A U = A

It is easy to interpret this law. Only the elements in "A" are common to universal set. Hence, intersection, being the set of common elements, is set "A".

Law of empty set

Since empty set is element of all other sets, it emerges that intersection of an empty set with any set is an empty set (empty set is only common element between two sets).

φ A = φ

Commutative law

The order of sets around intersection operator does not change the intersection. Hence, commutative property holds in the case of intersection operation.

A B = B A

Associative law

The associative property holds with respect to intersection operator.

A B C = A B C

The intersection of sets “A” and “B” on Venn’s diagram is :

Intersection of two sets

The intersection is a set of common elements and shown as colored region.

In turn, the intersection of set “A B” and set “C” is the small region in the center :

Intersection inloving three sets

Intersection of a set with "the intersection set of two sets"

It is easy to visualize that the ultimate intersection is independent of the sequence of operation.

Distributive law

The intersection operator( ) is distributed over union operator ( ) :

A B C = A B A C

We can check out this relation with the help of Venn diagram. For convenience, we have not shown the universal set. In the first diagram on the left, the colored region shows the union of sets “B” and “C” ie. B C . The colored region in the second diagram on the right shows the intersection of set “A” with the union obtained in the first diagram i.e. B C .

Distributive law

Distribution of intersection operator over union operator

We can now interpret the colored region in the second diagram from the point of view of expression on the right hand side of the equation :

A B C = A B A C

The colored region is indeed the union of two intersections : " A B " and " A C " . Thus, we conclude that distributive property holds for "intersection operator over union operator".

In the same manner, we can prove distribution of “union operator over intersection operator” :

A B C = A B A C

Analytical proof

Distributive properties are important and used for practical application. In this section, we shall prove the same in analytical manner. For this, let us consider an arbitrary element “x”, which belongs to set " A B C " :

x A B C

Then, by definition of intersection :

x A a n d x B C

x A a n d x B o r x C

x A a n d x B o r x A a n d x C

x A B o r x A C

x A B o r A C

x A B A C

But, we had started with " A B C " and used its definition to show that “x” belongs to another set. It means that the other set consists of the elements of the first set – at the least. Thus,

A B C A B A C

Similarly, we can start with " A B A C " and reach the conclusion that :

A B A C A B C

If sets are subsets of each other, then they are equal. Hence,

A B C = A B A C

Proceeding in the same manner, we can also prove other distributive property of “union operator over intersection operator” :

A B C = A B A C

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
Berger describes sociologists as concerned with
Mueller Reply
What is power set
Satyabrata Reply
Period of sin^6 3x+ cos^6 3x
Sneha Reply
Period of sin^6 3x+ cos^6 3x
Sneha Reply

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Source:  OpenStax, Functions. OpenStax CNX. Sep 23, 2008 Download for free at http://cnx.org/content/col10464/1.64
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